N = [ValueAtLF(-1), ValueAtLF(1)]
P = monomials(0:1, -1 .. 1)
M = [n(p) for p ∈ P, n ∈ N]
L11 = inv(M) * P
fplot(L11...)6 Shape functions and finite elements
6.1 Shape functions
Linear shape functions
Hermite polynomials
h = 2.0
N = [ValueAtLF(-1), DerivativeAtLF(-1), ValueAtLF(1), DerivativeAtLF(1)]
P = monomials(0:length(N)-1, IHat)
M = [n(p) for p in P, n in N]
H1 = inv(M) * P
fplot(H1...)Hermite like polynomials with middle node
h = 2.0
N = [ValueAtLF.(0:h/2:h); DerivativeAtLF.([0, h])] |> vcat
P = monomials(0:length(N)-1, 0 .. h)
M = [n(p) for p in P, n in N]
H = inv(M) * P
fplot(H...)Bilinear shape functions
WGLMakie.activate!()
V = [-1 1 1 -1; -1 -1 1 1]
N = [ValueAtLF(p) for p in eachcol(V)]
P = mmonomials(2, 1, QHat)
M = [n(p) for p in P, n in N]
L21 = inv(M) * P
fplot3d(L21)